Goals from Akos Buzsaky, Jay Simpson, Rowan Vine and Patrick Agyemang condemned Reading to a third consecutive defeat and left many of their fans at Loftus Road chanting for Rodgers to be sacked.
Rangers, who thrashed Preston 4-0 on Saturday, have scored 13 goals in their last three home matches and dominated this game despite Ben Watson being sent off for the second time in succession.
Watson - back from a ban after being dismissed at Swansea - had already been booked for an early foul when he took a free-kick before referee Andy Hall had blown the whistle.
Hall deemed that offence worthy of a second yellow card, and Rangers were reduced to 10 men.
But it was Reading's misfortune that the set-piece was not taken quickly, because as Watson was walking down the tunnel, Buzsaky curled the free-kick into the bottom corner of the net.
Rangers went further ahead six minutes before the break, when Wayne Routledge charging down the right and crossed to the far post, where Simpson netted his fourth goal since joining on loan from Arsenal.
Reading were unlucky not to reduce the deficit on the stroke of half-time, when Simon Church's free-kick struck the post.
But the Royals' hopes of a comeback were crushed by Ivar Ingimarsson's sending-off just 23 seconds into the second half.
The Icelander, booked in the first period, foolishly clattered into Simpson and was shown another yellow card.
Vine twice failed to finish after being set up by Routledge, but he made amends by firing home from the edge of the penalty area on 71 minutes.
And Agyemang added a fourth by racing onto Gavin Mahon's pass and rounding keeper Adam Federici before tapping into an empty net.
Reading were thoroughly outclassed, but did net a consolation goal when Brian Howard scored from close range with four minutes left.